Ocean Acoustic Problems Research Articles

Tank experiment: The ultrasonic approach to ocean physics

Acoustic/elastic waves ranging from a few kHz to a few MHz are nowadays easy to emit/receive through the use of "key in hands" multi‐channel systems. These systems provide the instantaneous amplitude and phase of the deterministic wave propagating in the medium with a dynamic larger than 90 dB in some cases (16‐bit amplitude sampling). The use of a large number of channels (at least 64) is mandatory to simultaneously investigate the spatial and temporal aspect of wave propagation in complex media. The advantage of a laboratory‐scaled ocean model relies in the ease with which it can be build, modified, and/or controlled over time. Finally, the trend in ocean wave physics is now to study the dynamics of such medium (internal waves, turbulent flow, sensor motion) that requires the use of a real‐time acquisition system. Examples of the types of analog experiments that could be carried out with a laboratory‐scaled system are the study of nonuniform doppler shifted fields in reverberant environments, ocean acoustic tomography and similar inverse problems, coherent communications in complex moving environments, etc. After the description of the equipment, we give more specific examples of potential experiments to illustrate the versatility of the laboratory set‐up.

An investigation into the validity of the Born approximation in reverberation calculations

Ocean reverberation models for active sonar applications assume the validity of the Born or single scatter approximation. In many areas of physics, a Born series does not converge due to a combination of the scattering strength and the spatial scale over which the scattering occurs. In these cases, the validity of the Born approximation is at best questionable. It is natural to attempt to examine the validity of the Born approximation and the convergence of the Born series in general for well-defined scattering problems in ocean acoustics. The problems examined in this study involve scattering from specific realizations of bottom roughness for 2-D waveguides. This study examines several idealized problems from the 1st Reverberation Workshop in November 2006 for the purpose of comparing a Born approximation solution to an exact solution provided by a 2-way coupled-mode integral equation approach. The coupled integral equations are solved exactly to within the limitations of finite differences and numerical quadrature. The exact solutions are then compared to those that solve the fundamental integral equation using the Born approximation and to solutions obtained with a finite-element model. [Work supported by ARL:UT IR&D.]

Efficient modeling of range-dependent seismo-acoustics problems

The parabolic equation method is the most effective approach for solving range-dependent problems in ocean acoustics. During the past three decades, many difficulties have been encountered while attempting to generalize this approach to problems involving elastic layers. With the development of an improved single-scattering solution [J. Acoust. Soc. Am. 121, 808813 (2007)], a large class of range-dependent seismo-acoustics problems can now be solved accurately and efficiently. Previous single-scattering solutions for seismic problems were based on an iteration formula, and limited by convergence problems. Since the improved solution does not require iteration, it provides the efficiency of the energy-conserving solution that has proved to be very useful for acoustics problems. The application of this approach to problems involving fluid layers is currently being investigated. [Work supported by ONR.]

Benchmarking range-dependent seismo-acoustic propagation problems

Developing the capability to accurately solve range-dependent problems in ocean acoustics has been going on for many years. A milestone in this development was the ASA benchmark effort in 1990 [J. Acoust. Soc. Am. 87, 1497–1545 (1990)], where a set of models was applied to generic range-dependent acoustic fluid problems to establish a set of benchmark solutions for model verification purposes. A further benchmarking effort, which included shear waves in the bottom, was published in 1997 [IEEE J. Ocean. Engn. 22, 226–236 (1997)] involving two seismo-acoustic codes based on spectral-integral techniques and a boundary-element code. Other codes have become available since then for solving general range-dependent elastic problems, either based on marching schemes that ignore backscattering (or adopt a single-scatter approximation) or techniques that solve the full boundary-value problem. In an attempt to establish a set of benchmark solutions for seismo-acoustic problems, we solve several low-frequency elastic problems involving changing bathymetry in multiple layers, and coupling between various wave types, including interface waves. The models in use or used are a parabolic equation model (RAM) and a commercial finite-element model (FEMLAB).

The resolution of a nonlinear ocean acoustic inverse problem

For a strictly linear inverse problem the model resolution matrix is a simple product of the linear system matrix and its generalized inverse. How much the model resolution differs from the identity is an indication of bias in the inverted model. Nonlinear inverse problems lead to higher order resolution operators, which allow one to quantify the effect of ignoring the nonlinearity in the inverse problem. In addition, the trade-off between resolution and variance is examined for the nonlinear inverse problem. Ocean acoustic inverse problems for ocean and/or bottom structure are intrinsically underdetermined, because some continuous quantity such as sound speed or temperature is estimated from a finite number of data values, and the model is generally discretely parametrized to be numerically tractable. Discretizing the model has the positive effect of regularizing an intrinsically ill-posed problem. However, discretizing an intrinsically nonlinear problem extracts a penalty. Higher order resolution operators may contribute even when they should be identically zero. (For example, in a quadratically nonlinear model, cubic and higher order resolution operators should be zero, but won’t be zero as a result of the discretization.) Modal propagation in an ocean acoustic waveguide is used to illustrate the issues. [Work supported by ONR.]

An analytic multiple frequency adjoint-based inversion algorithm for parabolic-type approximations in ocean acoustics

Inverse problems in ocean acoustics are generally solved by means of matched-field inversion in combination with meta-heuristic global search algorithms. Recently adjoint-based optimal nonlocal boundary control has been proposed as a possible complement or alternative to the traditional geoacoustic inversion methods. Especially for the assessment of a shallow water environment where field data are recorded on non-fully populated hydrophone arrays there is an inherent need for observation over a broad range of frequencies, as has been widely experimentally demonstrated. This article presents an analytic multiple frequency adjoint formulation for estimating an acoustically equivalent model of the seabed based on local impedance boundary conditions. It is intended as a benchmark solution for an extension of the approach to more sophisticated boundary conditions by means of automatic differentiation. The approach, coming from control theory, aims at inverting the impedance boundary condition along the water–sediment interface via measurements of the acoustic propagation in the water column. The impedance boundary condition thus plays the role of the control parameter of the complex pressure field in the waveguide. The developed multiple frequency adjoint method of optimal control provides an explicit representation of the gradient of the matched-field oriented cost function, and the article further proposes an analytical second-order adjoint formulation which permits the computation of the product of the Hessian matrix by a vector. By means of example inversion results the feasibility of the multi-frequency approach is discussed and the effect of a limited set of receiving locations is illustrated.

Validity of the Markov approximation in ocean acoustics

Moment equations and path integrals for wave propagation in random media have been applied to many ocean acoustics problems. Both these techniques make use of the Markov approximation. The expansion parameter, which must be less than one for the Markov approximation to be valid, is the subject of this paper. There is a standard parameter (the Kubo number) which various authors have shown to be sufficient. Fourth moment equations have been successfully used to predict the experimentally measured frequency spectrum of intensity in the mid-ocean acoustic transmission experiment (MATE). Yet, in spite of this success, the Kubo number is greater than 1 for the measured index of refraction variability for MATE, arriving at a contradiction. Here, that contradiction is resolved by showing that the Kubo parameter is far too pessimistic for the ocean case. Using the methodology of van Kampen, another parameter is found which appears to be both necessary and sufficient, and is much smaller than the Kubo number when phase fluctuations are dominated by large scales in the medium. This parameter is shown to be small for the experimental regime of MATE, justifying the applications of the moment equations to that experiment.

Range-dependent seismo-acoustics benchmark problems

Developing the capability to accurately solve range-dependent problems in ocean acoustics has been a topic of great interest for many years. At the time of the ASA benchmark efforts [J. Acoust. Soc. Am. 87, 1497–1545 (1990)], the elastic parabolic equation was in its early stages of development, and it was too early to consider problems involving shear waves. In recent years, there has been much progress in the development of parabolic equation and spectral techniques for range-dependent seismo-acoustics problems. The time is now ripe for extending the ASA benchmarks to cases involving shear waves. Several types of problems will be considered, including problems involving upslopes, downslopes, multiple solid layers, mode cutoff, and interface waves. [Work supported by the ONR.]

Assessing uncertainty in ocean acoustic inverse problems

The goal of solving ocean acoustic inverse problems is not just to find a model fitting the data. We can always fit an N-1th order polynomial to N data points, but it does not tell us much about the uniqueness, or limits placed on resolution in the presence of noise. The real goal is to develop more information than just a model that fits the data. For truly linear problems, constraints on the model information and uniqueness are characterized by the null space of the operator which maps the model to the data, the statistics are Gaussian, and the resolution matrices have well-defined meanings. Unfortunately most inverse problems of interest in ocean acoustics are nonlinear. We are faced with methods such as simulated annealing or genetic algorithms, which tell us nothing about the statistics and little about the uniqueness of our model solution, or exhaustive but numerically intensive Monte Carlo analyses. As an alternative we have examined a nonlinear filter, that is an extension of the Kalman and extended Kalman filters. We discuss the background of the ocean acoustic inverse problem for bottom properties, and how it can be addressed by employing an exact nonlinear filter. [Work supported by ONR.]

Adapting a truly nonlinear filter to the ocean acoustic inverse problem

Nonlinear inverse problems including the ocean acoustic problem have been solved by Monte Carlo, locally-linear, and filter based techniques such as the Extended Kalman Filter (EKF). While these techniques do provide statistical information about the solution (e.g., mean and variance), each suffers from inherent limitations in their approach to nonlinear problems. Monte Carlo techniques are expensive to compute and do not contribute to intuitive interpretation of a problem, and locally-linear techniques (including the EKF) are limited by the multimodal objective landscape of nonlinear problems. A truly nonlinear filter, based on recent work in nonlinear tracking, estimates state information for a nonlinear problem in continual measurement updates and is adapted to solving nonlinear inverse problems. Additional terms derived from the system’s state PDF are added to the mean and covariance of the solution to address the nonlinearities of the problem, and overall the technique offers improved performance in nonlinear inversion. [Work supported by ONR.]

Application of the Darboux Transform to Spectral Problems in Ocean Acoustics

We propose a generalization of the n-order Darboux transform, which can be applied to solving spectral problems in ocean acoustics in cases when the medium can be considered stratified with respect to density. For a second-order Darboux transform with the boundary conditions characteristic of a single-layer shallow sea model, conditions are determined under which the spectra of the initial and transformed spectral problems coincide. It is demonstrated that, within the framework of this model, it is possible to use the n-order Darboux transform for constructing sound velocity profiles admitting exact solutions.

A single-scattering solution that handles large contrasts across interfaces

Single-scattering and energy-conservation approximations have both proven to be effective for solving range-dependent ocean acoustics problems. The energy-conservation approach is usually used in acoustic models since it provides greater efficiency. The single-scattering approach is more general and more promising for problems involving elastic layers. A range-dependent medium is approximated in terms of a series of range-independent regions separated by vertical interfaces. The single-scattering solution is usually implemented in terms of an iteration formula, which may diverge when there is a large contrast in the material properties across an interface. Convergence can be improved by artificially splitting a vertical interface into a series of slices with smaller contrasts. For the idealized problem of scattering from a single stair step, a significant error may be introduced by neglecting multiple scattering between slices. However, the approach should be accurate for problems involving gradual range dependence, such as sloping interfaces. [Work supported by the Office of Naval Research.]

Helmholtz and parabolic equation solutions to a benchmark problem in ocean acoustics.

The Helmholtz equation (HE) describes wave propagation in applications such as acoustics and electromagnetics. For realistic problems, solving the HE is often too expensive. Instead, approximations like the parabolic wave equation (PE) are used. For low-frequency shallow-water environments, one persistent problem is to assess the accuracy of the PE model. In this work, a recently developed HE solver that can handle a smoothly varying bathymetry, variable material properties, and layered materials, is used for an investigation of the errors in PE solutions. In the HE solver, a preconditioned Krylov subspace method is applied to the discretized equations. The preconditioner combines domain decomposition and fast transform techniques. A benchmark problem with upslope-downslope propagation over a penetrable lossy seamount is solved. The numerical experiments show that, for the same bathymetry, a soft and slow bottom gives very similar HE and PE solutions, whereas the PE model is far from accurate for a hard and fast bottom. A first attempt to estimate the error is made by computing the relative deviation from the energy balance for the PE solution. This measure gives an indication of the magnitude of the error, but cannot be used as a strict error bound.

Frechet derivatives for shallow water ocean acoustic inverse problems

For any inverse problem, finding a model fitting the data is only half the problem. Most inverse problems of interest in ocean acoustics yield nonunique model solutions, and involve inevitable trade‐offs between model and data resolution and variance. Problems of uniqueness and resolution and variance trade‐offs can be addressed by examining the Frechet derivatives of the model‐data functional with respect to the model variables. Tarantola [Inverse Problem Theory (Elsevier, Amsterdam, 1987), p. 613] published analytical formulas for the basic derivatives, e.g., derivatives of pressure with respect to elastic moduli and density. Other derivatives of interest, such as the derivative of transmission loss with respect to attenuation, can be easily constructed using the chain rule. For a range independent medium the analytical formulas involve only the Green’s function and the vertical derivative of the Green’s function for the medium. A crucial advantage of the analytical formulas for the Frechet derivatives ...

A two-way integral equation approach to a shelf break problem

Benchmark solutions play an important role in the development, testing, and application of numerical models for range-dependent problems in underwater acoustics. To produce benchmark accuracy propagation solutions requires careful attention to both the theoretical formalism and numerical implementation. The focus of this research is a ‘‘benchmark’’ solution for a two-dimensional (2-D) shelf-break problem in ocean acoustics. The method employed to construct a solution is a two-way energy-conserving coupled mode approach in the form of coupled inhomogeneous integral equations. Solutions of the coupled equations are obtained using a powerful approach originally introduced in nuclear theory and also used to solve simple nonseparable problems in underwater acoustics. The basic integral equations are slightly modified to permit a Lanczos expansion to form a solution. The solution of the original set of integral equations is then easily recovered from the solution of the modified equations. Benchmark quality demands that the errors in each step of the numerical calculation be well quantified, so that the resolution in the transmission loss can be determined. Consistency checks are made in the form of using different numerical schemes to compute the adiabatic or free propagator and mode coupling matrix operator.

High-frequency propagation modeling with the parabolic equation

It is generally believed that the parabolic equation method rapidly becomes impractical above a few hundred Hertz for ocean acoustics problems. Results will be presented to illustrate that the split-step Padé algorithm [J. Acoust. Soc. Am. 100, 178–182 (1996)] is practical out to ranges of tens of kilometers for frequencies up to at least on the order of 10 kHz. The presentation will include a discussion of modifications to the parabolic equation model to handle effects that are usually neglected at lower frequencies, such as rough surfaces, internal waves, and volume attenuation in the water column. The examples will involve propagation in surface ducts and interaction with the seafloor. [Work supported by the Office of Naval Research.]

Forward and inverse modeling in porous media

Techniques for solving forward and inverse wave propagation problems involving poroelastic layers will be discussed. Parabolic equation techniques are efficient for solving problems in laterally varying media. A parabolic equation for poroelastic media has been developed and applied to problems in ocean acoustics [J. Acoust. Soc. Am. 98, 1645–1656 (1995)]. This approach has been generalized to the anisotropic case. The smallness of shear speeds in many ocean sediments has motivated the study of poroacoustic media [J. Acoust. Soc. Am. 104, 783–790 (1998)], which is a limiting case of Biot theory in which the rigidity vanishes. Although parabolic equation techniques have not been fully generalized from acoustics to poroelasticity, studying the intermediate case of poroacoustics has helped to bridge the gap. The parabolic equation techniques have been used as tools for solving inverse problems. This approach is presently being applied to field data. One of the issues that arises in solving the inverse problem is the mapping between the coefficients of the wave equation and the wave speeds. The inverse of this mapping can be used to define problems in terms of natural parameters rather than moduli. [Work supported by ONR.]

Estimating elastic sediment properties with the self-starter

The self-starter is implemented by the method of undetermined coefficients for ocean acoustics problems involving elastic sediment layers. This approach is efficient when the ocean bottom can be approximated by a relatively small number of homogeneous layers. For many problems, the self-starter only requires on the order of 10 wave number samples to provide accurate solutions at ranges on the order of 10 wavelengths. The geoacoustic inverse problem can be solved in nearly real time with this approach, which can generate on the order of a thousand forward solutions per second on the current generation of desk-top computers.

A review of recent results on ocean acoustic wave propagation in random media: basin scales

Measurements of basin-scale acoustic transmissions made during the last four years by the Acoustic Thermometry of Ocean Climate (ATOC) program have allowed for the study of acoustic fluctuations of low-frequency pulse propagation at ranges of 1000 to 5000 km. Analysis of data from the ATOC Acoustic Engineering Test conducted in November 1994 has revealed new and unexpected results for the physics of ocean acoustic wave propagation in random media. In particular, use of traditional /spl Lambda/, /spl Phi/ methods (using the Garrett-Munk (GM) internal wave model) to identify the wave propagation regime for early identifiable wavefronts predict the saturated regime, whereas observations of intensity probability density functions, intensity variance, and pulse time spread and wander suggest that the propagation is more likely near the border between the unsaturated and partially saturated regimes. Calculations of the diffraction parameter /spl Lambda/ are very sensitive to the broad-band nature of the transmitted pulse, with CW calculations differing from a simplistic broad-band calculation by 10/sup 3/. A simple model of pulse propagation using the Born approximation shows that CW and broad-band cases are sensitive to a random medium very differently and a theoretical description of broad-band effects for pulse propagation through a random media remains a fundamental unsolved problem in ocean acoustics. The observations show that, at 75-Hz center frequency, acoustic normal mode propagation is strongly nonadiabatic due to random media effects caused by internal waves. Simulations at a lower frequency of 28 Hz suggest that the first few modes might be treated adiabatically even in a random ocean. This raises the possibility of using modal techniques for ocean acoustic tomography, thereby increasing the vertical resolution of thermometry. Finally, the observation of unsaturated or partially saturated propagation for 75-Hz broad-band transmissions, like those of ATOC, suggests that ray-based tomography will be robust at basin-scales. This opens up the possibility of ray-based internal wave tomography using the observables of travel time variance, and vertical and temporal coherence. Using geometrical optics and the GM internal wave spectrum, internal wave tomography for an assortment of parameters of the chi model can be formulated in terms of a mixed linear/nonlinear inverse. This is a significant improvement upon a Monte Carlo approach presented in this paper which is used to infer average internal wave energies as a function of depth for the SLICE89 experiment. However, this Monte Carlo approach demonstrated, for the SLICE89 experiment, that the GM model failed to render a consistent inverse for acoustic energy which sampled the upper 100 m of the ocean. Until a new theory for the forward problem is advanced, internal wave tomography utilizing the signal from strong mode coupling can only be carried out using time-consuming Monte Carlo methods.

Theory of sound propagation in the ocean: A tribute to L. M. Brekhovskikh

This presentation will be an account (from a personal perspective) of the impact L. M. Brekhovskikh’s first (and now classic) book on Waves in Layered Media from 1960 has had on the development of modern-day numerical models for solving ocean acoustic problems. A whole generation of acoustic modelers, starting in the early 1970s, devoted a career to developing computer codes that would predict sound propagation through a realistic ocean environment, including refraction, scattering, and diffraction of sound within the water column itself, as well as reflection and scattering at ocean boundaries (sea surface and seabed). Not everybody realized that much of the underlying theory had been developed years before and presented in the monograph by Brekhovskikh in 1960 (English edition). In fact, much of the numerical work undertaken in the 1970s and 1980s was merely an automation—through the use of a computer—of solution procedures already outlined by Brekhovskikh. Few, if any, have contributed more to establishing a sound theoretical foundation for ocean acoustics, and L. M. Brekhovskikh’s influence on the field can hardly be overestimated.